Breaking the mirror symmetry of photonic spin-orbit interaction using a geometrically symmetric chiral resonator (2024)

Photonic spin-orbit interaction (PSOI) encompasses the coupling of intrinsic spin angular momentum (SAM, associated with circular polarization) and extrinsic orbital angular momentum (OAM) of light [1–11], allowing for the manipulation of light propagation through interacting with artificial structures like waveguides and metasurfaces [5, 12–17]. Importantly, the polarization information encoded in SAM can be converted into OAM-encoded information of light and vice versa. Such conversion results in many exotic spatial distribution profiles of light fields, such as optical vortices, twisted light, and unidirectional transmission or spatial spin density distributions [1, 2], significantly enriching the toolbox of all-optical modulation of light. In optical communication, cylindrical waveguide modes sustained in optical fibers are utilized to transport photons as information carriers, confining them in wavelength-scale mode volumes within the fibers. When the diameter of an optical fiber reduces to the submicrometer scale [18], a significant amount of evanescent fields emerge and the guided modes inside/outside the air-clad nanofiber surface exhibit a transverse SAM with its direction locked to the propagation direction (i.e. the direction of the extrinsic OAM) of the guided modes [19, 20]. Similar transverse spin-direction locking due to PSOI has also been observed in other waveguide structures, including rectangular dielectric waveguides and photonic crystal waveguides [21–23]. This is an inherent property of evanescent waves and thus can be extended from dielectric waveguides to surface plasmonic metal waveguides [24–26].

Photonic waveguides including nanofibers typically exhibit high geometric symmetries, such as inversion symmetry, mirror symmetry, and rotational symmetry which manifest in their transverse spin profiles. For instance, the mirror symmetry of a nanofiber dictates that its transverse spin undergoes a flip while maintaining the same density for radially opposite positions [27]. This intrinsic symmetry results in equivalent coupling directionality towards the two ends of the nanofiber when employing enantiomeric circularly-polarized emitters. Placing a circularly polarized emitter at radially opposite positions leads to a flip in the coupling direction, enabling high degree of operational freedom. However, this means the degree of circular polarization and the accuracy of the excitation spot location/size, respectively, are critical. By intentionally breaking the symmetries of nanofiber, the unidirectional transportation of light can be achieved with reduced requirements. A prior study demonstrated that an anisotropic etched metal groove, breaking the inversion and rotational symmetry of the corresponding surface plasmon polariton groove waveguide, can be arranged side by side as a metasurface device to generate unidirectional transportation of valley-polarized excitons in transition metal dichalcogenides regardless of the beam-size/location-sensitivity of PSOI interfaces [28]. Additionally, this inversion and rotational symmetry can be broken by introducing a nanosphere/'U'-shape waveguide as the scatter/coupler sitting transversely to waveguide, directing the light with flexible excitation beam size and position [23, 29].

In this paper, we propose a novel approach to break the mirror symmetry of PSOI interface by placing a chiral gold nanohelicoid (GNH) resonator on a nanofiber. The GNH, modelled based on a recently developed synthesis method, features a geometrically symmetric geometry with thirteen rotational axes and one inversion centre inherited from its cube geometry while the mirror symmetry is broken due to its fine surface structure [30, 31]. In comparison with employing a symmetric nanocube as the scatter, this configuration avoids one C₂ rotational symmetric axis while breaking mirror symmetry and inverse symmetry, resulting in robust PSOI unaffected by the beam size and excitation polarization. Finite-difference time-domain (FDTD) calculations reveal that, with left-circularly polarized (LCP)/right-circularly polarized (RCP) excitation, unidirectional light transportation manifests differently in a single mode fiber. Consequently, linearly polarized light, comprising equal magnitude orthogonal circular polarization states with a fixed phase difference, can be harnessed for achieving unidirectional light transportation. This extension broadens the potential applications of PSOI interfaces to polarization-insensitive devices. Most notably, the far-field circular dichroism (CD) of the chiral GNH is projected to the directionality of PSOI interface through chiral near-field spin density, proposing the possibility of measuring plasmonic/molecular CD through linear polarization light-excited PSOI interfaces or actively controlling directionality through wavelength-selected excitation.

The purposed GNH-nanofiber hybrid system consists of a GNH positioned on a single-mode silica nanofiber with a diameter (figure 1(a)). This configuration is expected to be readily achieved through the random drop-casting of GNH, ensuring practical feasibility. The GNH serves as a scatter that couples plane wave excitation into the nanofiber guide mode. Numerical simulations using three-dimensional (Lumerical FDTD) method with a finite-difference eigenmode (FDE) solver are employed to reveal the mode profile and spin density profile of the nanofiber. The nanofiber's mode profile (figure 1(b)) illustrates the electric field distribution, with the nanofiber demarcated by the white dashed line. Degenerate modes with different polarizations coexist in the nanofiber. However, for excitation with wavevector k in the z direction and polarization in the x-y plane, the y-axis-polarized mode is significantly coupled in the nanofiber. Beyond the nanofiber, a pronounced evanescent field is observed, contributing to elevated photonic state and spin densities, thereby inducing significant PSOI. The nanofiber, with a diameter of 400 nm, exhibits a single guide mode at 600 nm, which matches with the resonance wavelength of the GNH. The selection of the diameter of the nanofiber supporting single-mode at 600 nm is based on a set of numerical simulations involving variations in the diameter of the nanofiber. In figure 1(c), the spin density profile of the nanofiber mode in the transverse cross-section (y-z plane) is determined using the local Stokes parameter S₃ = −2Im(E_xE_y *)/(|E_x |²+|E_y |²). The transverse spin density in z direction is calculated using the x/y components of the electric field to overlap with the SAM of the planewave excitation which have a k vector in z direction. Notably, nearly-unity transverse spin density with spin flipping for radially opposite positions is evident inside and outside the nanofiber, while at the symmetric plane (y = 0), spin density is zero. The electric field distribution profile and the spin density profile of the longitudinal plane (the dashed orange line in figure 1(a), i.e. z = 0 plane) are shown in figures 1(d) and (e). The fields are evenly distributed across longitudinal direction for a smooth nanofiber. High symmetry is obtained for bare nanofiber, as illustrated by figures 1(b)–(e), indicating that same amount of spin density is obtained for radially opposite positions. This observation suggests that if a plane wave is excited onto bare nanofiber, no unidirectional transportation of light will be realized. However, when a local chiral dipole or scatter is placed near the nanofiber, the transportation direction becomes dependent on the handedness and location of the chiral dipole (scatter), indicating the stringent requirements of excitation conditions to realize unidirectional transportation. It is essential to note that multimode nanofibers also exhibit symmetric PSOI and large local spin density, as detailed in our previous publication [27].

The schematic representation of the chiral GNH and its high-symmetric geometrical point are illustrated in figure 2(a). Its geometrical chirality is achieved by breaking mirror symmetry at the edges of a nanocube. The rotational and inversion symmetry of the GNH is well preserved as it has three C₄ axes, four C₃ axes, six C₂ axes and one inversion centre, inherited from its cube-like geometry (figure 2(b)). Unlike conventional anisotropic chiral nanoparticles like helices, the GNH is high-symmetric, facilitating its integration into photonic structures without intricate placement considerations. Different size of the series of geometrically symmetric chiral GNHs, along with their enantiomers, can be synthesised using a seed-mediated growth method with the guidance of L-/D-cysteine [30, 31], generating significant interest among diverse audiences so far [32–34]. For this study, the. Edge length of the GNHs model is 100 nm. To investigate its chiral optical resonance, Lumerical FDTD is employed to calculate the extinction cross-section and corresponding CD as illustrated in figure 2(c). LCP and RCP excitations are modeled by two perpendicular total-field scattered-field (TSTF) light sources with phase differences of 90 degree, respectively. Similar to a gold nanocube with the same edge length, the chiral GNH exhibits a broad plasmonic resonance across 550–700 nm, with absorption and scattering peaks at 567 nm and 625 nm, respectively (depicted by the black line in figure 2(c)). The CD spectrum is calculated by discerning the difference in extinction cross-section under LCP and RCP excitations. It shows a Fano-like feature with a dip and peak at 525 nm and 581 nm respectively due to the Cotton effect [35]. This observation attests to the polarization dependence and optical chirality of the chiral GNH in the far-field, demonstrating its intriguing optical properties.

The PSOI of the hybrid interface is numerically demonstrated by applying RCP/LCP excitations in the −z direction and monitoring the transmission towards two directions of the nanofiber, donated as T_L and T_R, respectively (see figure 3(a)). The total coupling efficiency is represented by the sum of the transmission intensities T_L + T_R. The calculated result is depicted as the red line in figure 3(b). A broad resonance peak is observed from 570 nm to 800 nm, indicating a red-shifted chiral GNH plasmonic resonance peak with a dominating scattering component, confirming that the coupled light originates from the scattering of the GNH. The directionality of the interface (D), defined as (T_R−T_L)/(T_R + T_L), is calculated to quantify the degree of unidirectional transportation. Across the resonance region, D ∼ 0.5 is obtained, signifying opposite directions of light transmission into the nanofiber. It is noteworthy that, in contrast to a symmetric system, such as a nanocube on a nanofiber, where the directionality for LCP and RCP excitations is symmetric with flipped signs, for the chiral GNH, the absolute values of the directionality are asymmetric D_LCP + D_RCP ≠ 0, as shown in the lighter blue line in the insert of figure 3(b). This is due to the asymmetric spin density induced in chiral GNH [31, 36], which further couples into nanofiber through PSOI. Notably, a Fano-like directionality spectrum is obtained with a dip and peak at 559 nm and 590 nm, corresponding to the CD spectrum of the chiral GNH with a slight red shift. This is particularly intriguing because CD spectrum indicates the energy difference of the scattering state in the far-field, while the directionality dichroism indicates spin density difference of the scattering state in the near-field. Typically, the far-field scattering efficiency is correlated to the near-field local electric field intensity. The same dichroism features in our calculation suggest a correlation between near-field chirality and far-field efficiency. Figures 3(c) and (d) illustrate the electrical field distribution on the longitudinal cross-section of the nanofiber (x-y plane). Depending on the circular polarization of the excitation plane wave, light is transmitted in opposite directions, clearly depicting a unidirectional transportation trajectory. To enhance clarity, local electrical field intensities are displayed in logarithmic scale at a wavelength of 590 nm.

The primary advantage of employing a mirror-symmetry-broken PSOI interface is the achievement of unidirectional transportation of light when a linearly polarized excitation is applied. Figure 4(a) provides an artistic illustration of PSOI at the chiral GNH-nanofiber interface under linearly polarized excitation. When a linearly polarized light beam illuminates the interface along the −z direction, as shown in the upper figure, the intrinsic near-field chirality of the GNH acts as a local director, causing light to couple to the fibre in a preferential direction. When the light beam is incident from the opposite direction, i.e. +z direction, the light trajectory flips because of the C₂ symmetry of the hybrid system. This expectation is confirmed by the numerical calculation results shown in figure 4(b). The transmission spectrum remains consistent with the chiral excitation scenario, and the directionality spectrum is symmetric for excitations with opposite k-directions. Notably, when the excitation is outward (+z direction), the directionality spectrum aligns with the absolute values of RCP and LCP directionality, exhibiting a dip and peak at 559 nm and 590 nm, respectively. This indicates that the coupling directions for linear excitations at 559 nm and 590 nm flip. This is consistent with our expectation because a linear excitation corresponds to the sum of two orthogonal circular polarization states, and the wavelength-dependent directionality paves the way for active control of spectral directionality. The directionality for a symmetric nanocube-nanofiber interface is zero, as indicated by the gray line. The electric field distribution for the inward (−z direction) excitation is calculated and plotted in figure 4(c). The 'wiggly' coupling patterns along the fiber are similar to that observed in figures 3(c) and (d), consistent with the combination of two orthogonal polarization states. Although unidirectional transportation is not discernible to the naked eye as clear as figure 4(a) due to the low directionality of 0.008, sophisticated designs of chiral nanoparticles could yield a substantial and unified chiral near-field distribution, enhancing the directionality [31, 35]. To clearly illustrate the behavior of unidirectional transportation, we compare the electric field distribution profiles under outward and inward 590 nm illumination and calculate the local field intensity difference between the two scenarios (log₁₀|E_+z |−log₁₀|E−z|@590 nm): as plotted in figure 4(d), the outward (+z direction) excitation is inclined to couple to the right side of the fibre (+x direction), as indicated by the red fiber mode trajectory, whereas the inward (−z direction) excitation prefers to couple to the left side of the fibre (−x direction), as indicated by the blue fiber mode trajectory.